Applied Mathematics, 2018, 9, 1238-1257 http://www.scirp.org/journal/am

ISSN Online: 2152-7393 ISSN Print: 2152-7385

DOI: 10.4236/am.2018.911081 Nov. 20, 2018 1238 Applied Mathematics

A Mathematical Modelling of the Effect of Treatment in the Control of Malaria in a Population with Infected Immigrants

Olaniyi S. Maliki1, Ngwu Romanus1, Bruno O. Onyemegbulem2

1Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria 2African Center of Excellence in Phytomedicine, Research and Development University of Jos, Jos, Nigeria

Abstract

In this work, we developed a compartmental bio-mathematical model to study the effect of treatment in the control of malaria in a population with infected immigrants. In particular, the vector-host population model consists of eleven variables, for which graphical profiles were provided to depict their individual variations with time. This was possible with the help of MathCAD software which implements the Runge-Kutta numerical algorithm to solve numerically the eleven differential equations representing the vector-host malaria population model. We computed the basic reproduction ratio 0R following the next generation matrix. This procedure converts a system of ordinary differential equations of a model of infectious disease dynamics to an operator that translates from one generation of infectious individuals to the next. We obtained 0 0 0m hR R R= × , i.e., the square root of the product of

the basic reproduction ratios for the mosquito and human populations re-spectively. 0mR explains the number of humans that one mosquito can in-fect through contact during the life time it survives as infectious. 0hR on the other hand describes the number of mosquitoes that are infected through contacts with the infectious human during infectious period. Sensitivity analysis was performed for the parameters of the model to help us know which parameters in particular have high impact on the disease transmission, in other words on the basic reproduction ratio 0R .

Keywords

Malaria Control, Infected Immigrants, Basic Reproduction Ratio, Differential Equations, MathCAD Simulation

How to cite this paper: Maliki, O.S., Ro-manus, N. and Onyemegbulem, B.O. (2018) A Mathematical Modelling of the Effect of Treatment in the Control of Malaria in a Population with Infected Immigrants. Ap-plied Mathematics, 9, 1238-1257. https://doi.org/10.4236/am.2018.911081 Received: July 23, 2018 Accepted: November 17, 2018 Published: November 20, 2018 Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

Open Access

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1. Introduction

Malaria is a highly prevalent infectious disease especially in the tropical and sub-tropical areas. Figure 1 below is a map obtained from WHO Malaria Report 2010 [1], depicting the countries where malaria was endemic in 2009 (shaded re-gion).

In addition to being widespread, malaria is also a deadly disease. This is be-cause statistics has shown that for Africa in particular, annually 145,000 million to 150,000 million infections are reported, among which, 800 to 850 cases result in deaths as shown in Table 1. Most of the deaths are either children under five or pregnant women. Typical symptoms of malaria infections start with head-ache, followed by periodic bouts of fevers and chills, and sometimes even coma. The period of cyclical fevers lasts several days, during which time a high proba-bility of dying has been observed for children, since their immune systems are weak. Such fever can also lead to abortions in pregnant women.

1.1. Brief Analysis of Malaria Data

It is interesting to do a quick statistical analysis of the data in Table 1, for the malaria cases in Africa as provided by WHO (Figure 2). We perform a nonli-near regression analysis for both the reported cases (C) and deaths (D) against time (T). The result follows from SPSS.

Figure 1. Malaria endemic countries 2009.

Table 1. Estimates of malaria cases and deaths in Africa by WHO, 2000-2009.

Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Cases (×103) 173,000 178,000 181,000 185,000 187,000 188,000 187,000 186,000 181,000 176,000

Deaths (×103) 900 893 885 880 870 853 832 802 756 709

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Figure 2. Quadratic regression model for malaria cases 2000-2009. Model Summary and Parameter Estimates. Dependent Variable: C (Numbers of cases)

Equation Model Summary Parameter Estimates

R Square F df1 df2 Sig. Constant b1 b2

Quadratic 0.981 180.044 2 7 0.000 165,283.333 7609.848 −647.727

The independent variable is T.

Observation: It is quite clear from the WHO data, for the number of malaria

cases reported over the 10 year period that the incidence of malaria infection follows a parabolic curve, rising sharply initially, to reach a maximum and then declining sharply thereafter (Figure 3). The equation of the parabola is given by:

2165283.3 7609.85 647.73C T T= + − with goodness of fit 2 0.981R = .

Model Summary and Parameter Estimates. Dependent Variable: D (Number of deaths)

Equation Model Summary Parameter Estimates

R Square F df1 df2 Sig. Constant b1 b2

Quadratic 0.992 438.638 2 7 0.000 882.883 12.070 −2.890

The independent variable is T.

Observation: The number of malaria related deaths over the 10 year period as

depicted in the above graph, follows a parabolic curve, rising from a high value initially, then reaching a maximum and then declining sharply thereafter. The equation of the parabola is given by:

2882.883 12.07 2.89D T T= + − with goodness of fit 2 0.992R = .

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Figure 3. Quadratic regression model for deaths caused by malaria 2000-2009.

1.2. Life Cycle of Malaria Parasites

Malaria is a vector-borne disease [2]. Malaria parasites are transferred between humans through mosquitoes. The malaria parasite life cycle is divided into two parts, one is within host (human) body and the other is within vector (mosquito) body.

Human infection starts from a blood meal of an infectious female mosquito. The parasites existing in the infectious mosquito’s saliva, called sporozoites at this stage, enter the bloodstream of the human through mosquito bites and mi-grate to the liver. Within minutes after entering in the human body, sporozoites infect hepatocytes, and multiply asexually and asymptomatically in liver cells for a period of 5 - 30 days [3]. This period is called the exo-erythrocytic stage. At the end of this stage, thousands of merozoites (schizonts) emerge inside an infected liver cell. These merozoites rupture their host cells undetectably by wrapping themselves in the membrane of infected liver cells. Then, merozoites escape into the bloodstream and get ready to infect red blood cells. Once entering the bloodstream, free merozoites undergo the so-called erythrocytic stage, in which merozoites invade red blood cells to develop ring forms before experiencing asexual or sexual maturation. Within the red blood cells, a proportion of para-sites keep multiplying asexually and periodically break out of infected old red blood cells to invade fresh red blood cells. Such amplification cycles may cause the symptom of waves of fever. The remaining parasites follow sexual matura-tion and produce male (micro-) and female (macro-) gametocytes which may be taken up by bites of female mosquitoes. Finally, when it has developed into an infectious form, it spreads the disease to a new mosquito that bites the infectious human.

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1.3. Malaria Control and Treatments

According to the transmission procedure of malaria, there are three conditions for the prevalence of the disease:

1) High density of Anopheles mosquitoes, 2) High density of human population, 3) Large rate of transmission of parasites between human beings and mosqui-

toes. Obviously, not too much can be done in respect to (2). So, (1) and (3) are na-

turally targeted. That is, either controlling the population of Anopheles female mosquitoes at a lower level, or avoiding biting by mosquitoes can reduce the chance of malaria becoming endemic. In the middle of the last century, people in Africa have already knew how to remove or poison the breeding grounds of mosquitoes or the aquatic habitats of the larva stages, such as by filling or ap-plying oil to places with standing water, to control the population of mosquitoes [4]. Later, pesticide was widely employed to eliminate mosquitoes. On the other hand, mosquito nets, bedclothes and mosquito-repellent incense (indoor resi-dual spraying) also help to keep mosquitoes far away from people and minimize the biting rate, greatly reducing the chance of infection and transmission of ma-laria. There are some effective drugs for malaria patients currently. For example, Chloroquine, Quinuine, Primaquine and combinations of some other drugs like sulfadoxine and pyrimethamine (SP) are effective medicines for treating infec-tions caused by the five major parasites. Although malaria is an entirely pre-ventable or curable disease thanks to these effective medicines, there are still millions of people suffering from this disease, who are too poor to afford full treatments. Moreover, insufficient treatments due to poor economic conditions, may result in drug resistance and lead to emergence of new (drug resistant) strains of malaria parasites. For instance, the first case of resistance to Chloro-quine was documented in 1957. Chloroquine, Quinine and Sulfadoxine-pyrime- thamine resistance cases have been reported in almost all disease endemic areas [5].

1.4. Control of Mosquito-Borne Infections

In order to control mosquito-borne infections one can adopt the following measures; Reduce vector population: Make environment less mosquito-friendly by

draining stagnant water. Use insecticides; not without problems: for example some mosquitoes be-

come insecticide resistant. Prevent mosquitoes biting people. Insecticide-laced bed nets, although this is

ineffective against mosquitoes that mainly bite during the day (e.g. A. aegyp-ti).

Vaccines and drug treatments. Not always available, there are problems with drugs and drug resistance.

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1.5. The Ross-Macdonald Malaria Model

The first and simplest model of malaria was developed by [6] Ross and later ex-tended by Macdonald [7]. This so-called Ross-Macdonald model is the best-known and most widely used model. Despite its simple structure as shown below, it enables us to interpret and compare a broad range of epidemiological models.

1.6. Remark

In the Ross-Macdonald model of malaria transmission, the flow of human from a susceptible class to an infected class and through recovery from infection, the reverse is shown in the upper part of the Figure 4. The flow of mosquitoes from susceptible class to an infected class, and finally to an infectious class is shown further down. The human and mosquito population are linked through the transmission process.

1.7. Statement of the Problem

The development of the means intended to reduce the spread of malaria infec-tions and eradication necessitates decisive measures to curb the malaria epidem-ic. In particular, sustained minimization of the number of humans with inci-dence of malaria as a result of adequate control, can be attained by developing a suitable mathematical model which can enable us to understand better the dy-namics and control of the vector-host endemic.

In developing the model, the human population is compartmentalized into seven classes including the susceptible, infected, exposed, treated, non-treated, recovered, and protected classes. For the mosquito population, we have four classes, namely; class of mosquito larva, susceptible mosquitoes, infected mos-quitoes and exposed mosquitoes. We assume free interaction between the vector and host populations. The mathematical analysis of the compartmental models leads us to eleven coupled systems of nonlinear ordinary differential equations.

2. Construction of the Compartmental Model

In this section we develop a compartmental bio-mathematical model (Figure 5)

Figure 4. The ross-macdonald malaria model.

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Figure 5. Compartmental model for human-mosquito interaction. to study the effect of treatment in the control of malaria in a population with in-fected immigrants.

From the above compartmental model we obtain the following equations for the dynamics of the human-mosquito interaction.

2.1. Human Population

( ) 2 1d 1d

H H M HH H H H H

H

S I Sq A R S St N

βα ρ α δ= − Λ + + − − − (1)

dd

H H M HH H H

H

E I S gE Et N

β δ= − − (2)

1 2dd

HH H H H H H

I gE q k I k I It

δ= + Λ − − − (3)

( )2d

dHN

H H H HNI k I I

tω δ= − + (4)

1dd

HH H H H

T k I T Tt

γ δ= − − (5)

( )dd

HH H H

R T Rt

γ µ ρ δ= − + + (6)

1 2dd H H HA S R A At

α µ α δ= + − − (7)

2.2. Mosquitoe Population dd

MM M M M

L mL Lt

δ= Λ − − (8)

dd

M M H MM M M

H

S I SmL St N

β δ= − − (9)

dd

M M H MM M M

H

E I S E Et N

β φ δ= − − (10)

dd

MM M M

I E It

φ δ= − (11)

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2.3. Remark

The state variables and parameters are defined in Table 2 and Table 3 respec-tively. Table 2. State variables of the basic malaria model.

Symbol Description

( )HS t Susceptible human population at time t

( )HE t Exposed human population at time t

( )HI t Infected human population at time t

( )HNI t Non-treated infected human population at time t

( )HT t Treated human population at time t

( )HR t Recovered human population at time t

( )A t Protected human population at time t

( )ML t Population of mosquito larva at time t

( )MS t Population of susceptible mosquitoes at time t

( )ME t Population of exposed mosquitoes at time t

( )MI t Population of infected mosquitoes at time t

HN

Total population size of humans

MN Total population size of mosquitoes

Table 3. Parameters of the basic malaria model.

Symbol Description

HΛ Birth and immigrant rate of humans

MΛ Birth rate of mosquitoes

ρ Rate of loss of immunity

Hβ Transmission rate of infection from infected mosquitoes to susceptible human

2α Loss of immunity of protected class

q Fraction of infective immigrants

1α Progression rate of susceptible human to protected class

1k Treatment rate of human from infected state to treated class

2k

Transmission rate of human from infected state to infectious none treated class

g Progression rate of human from exposed to infected compartments

γ Recovery rate of human from treated class

Hδ Natural death rate of human from exposed to infected

µ Progression rate of human from recovery class to protected class

M Progression rate of mosquitoes from larva to susceptible

Mβ Transmission rate of infection from infected human to susceptible mosquitoes

Mδ Natural death rate of mosquitoes

φ

Progression rate of exposed mosquitoes to infected mosquitoes

Hω Disease-induced death rate of human

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2.4. Invariant Region

The total population sizes HN and MN can be determined by

H H H H HN H HN S E I I T A R= + + + + + + and M M M M MN L S E I= + + + . Thus d

dH

H H H H HNN N I

tδ ω= Λ − − (12)

Without loss of generality, we can write

d d, d d

H MH H H M M M

N NN Nt t

δ δ≤ Λ − ≤ Λ − (13)

2.5. Lemma

The model system has solution which are contained in the feasible H MΩ = Ω ×Ω . Proof: let 11, , , , , , , , , ,H H H HN H H M M M MS E I I T A R L S E I +Ω = ∈ be any solu-

tion of the system with non-negative initial conditions. From Equation (13) d

dH

H H HN N

tδ≤ Λ − (14)

Adopting Birhoff and Rotta [8] theorem on differential inequality, we have

0 , e HtHH H H H

H

N N C δδδ

−Λ≤ ≤ Λ − ≥ (15)

where C is a constant. Therefore, all feasible solutions of the human population only of the model

system are in the region.

( ) 7, , , , , , : HH H H H HN H H H

H

S E I I T A R Nδ+

ΛΩ = ∈ ≤

Similarly the feasible set for model of the mosquitoes population only are in the region

( ) 4, , , : MM M M M M M

M

L S E I Nδ+

ΛΩ = ∈ ≤

Therefore the feasible set for the model system is given by

( ) 11 *

*

, , , , , , , , , , : ,HH H H HN H H M M M M H H

H

MM M

M

S E I I T A R L S E I N N

N N

δ

δ

+

ΛΩ = ∈ ≤ =

Λ

≤ =

(16)

2.6. Mathematical Analysis of the Model

The nonlinear system (1)-(11) will be qualitatively analyzed so as to find the conditions for existence and stability of disease free equilibrium points. Analysis of the model allows us to determine the impact of treatment on the transmission of malaria infection in a population. Also on finding the reproductive number

0R , one can determine if the disease become endemic in a population or not [9]. However, one can see that adding the human equation of the model, with the case that there is no disease -induced death. From Equation (13)

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dd

HH H H

N Nt

δ= Λ − , hence ( ) HH

H

N tδΛ

→ as t →∞ .

Thus H

HδΛ is the upper bound of ( )HN t provided that ( )0 H

HH

NδΛ

≤ . Si-milarly,

( )d d

M MM M M M

M

N N N tt

δδΛ

= Λ − ⇒ → as t →∞ .

Thus M

MδΛ is the upper bound of ( )MN t provided that ( )0 M

MM

NδΛ

≤ .

Hence the invariant region is

( ) 11 *

*

, , , , , , , , , , : ,HH H H HN H H M M M M H H

H

MM M

M

S E I I T A R L S E I N N

N N

δ

δ

+

ΛΩ = ∈ ≤ =

Λ

≤ =

is positively invariant. Hence no solution path leaves through and boundary of Ω . Since path cannot leave Ω , solution remains non-negative for non negative initial conditions. This means that the solution exists for all positive time t. Therefore the model (1)-(11) is mathematically and epidemiological well-posed [10].

For convenience and to simplify the analysis of our model, we rewrite the model system (1)-(11) in terms of the proportions of individual in each class. Let

, , , , , ,

, , , , .

HNH H H H Hh h h h h hn

H H H H H H

M M M Mm m m m

H H H H H

IS E I T Rs e i t r iN N N N N N

L S E IAz l s e iN N N N N

= = = = = =

= = = = =

Let M

H

NN

π = be the female mosquito–human ratio, that is, the number of

female mosquito per human host. The ratio M

H

NN

π = is constant because a

mosquito takes a fixed number of blood meals per unit independent of the pop-ulation density of the host [11]. Also let

, , , , , , .H h M m H h H h M m M m H hβ β δ δ β β δ δ ω ωΛ = Λ Λ = Λ = = = = =

The simplified model now becomes modified human and mosquito popula-tion models.

2.7. Modified Human Population

( ) 2 1d 1d

hh h h m h h h h

s q z r i s s st

α ρ β α δ= − Λ + + − − − (17)

dd

hh m h h h h

e i s ge et

β δ= − − (18)

1 2dd

hh h h h h h

i ge q k i k i it

δ= + Λ − − − (19)

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( )2dd

hnh h h hn

i k i it

ω δ= − + (20)

1dd

hh h h h

t k i t tt

γ δ= − − (21)

( )dd

hh h h

r t rt

γ µ ρ δ= − + + (22)

1 2dd h h hz s r z zt

α µ α δ= + − − (23)

2.8. Modified Mosquitoes Population

dd

mm m m m

l ml lt

δ= Λ − − (24)

dd

mm m h m m m

s ml i s st

β δ= − − (25)

dd

mm h m m m m

e i s e et

β φ δ= − − (26)

dd

mm m m

i e it

φ δ= − (27)

2.9. Positivity of Solutions

It is necessary to prove that all solutions of system (17)-(27) with positive initial data will remain positive for all times 0t > . This will be established by the fol-lowing theorem.

2.10. Theorem

Let the initial data be

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 0 0, 0 0, 0 0, 0 0, 0 0, 0 0,

0 0, 0 0, 0 0, 0 0, 0 0h h hn h h

h m m m m

s i i t z r

e s l e i

≥ ≥ ≥ ≥ ≥ ≥

≥ ≥ ≥ ≥ ≥ ∈Ω

Then the solution set ( ) ( ), , , , , , , , , ,h h h hn h h m m m ms e i i t z r l s e i t of the model sys-tem (4) is positive for all 0t > .

Proof: From first equation of (17)

( ) ( )2 1 1d 1d

hh h h m h h h h h m h h

s q z r i s s s i st

α ρ β α δ β α δ= − Λ + + − − − ≥ − + +

( ) ( )11 d dh h m hh

s i ts

β α δ⇒ ≥ − + +∫ ∫

( ) ( ) ( )1 0 e 0h m hi t

h hs t s β α δ− + +∴ ≥ ≥

Following the above procedure, from equations (18)-(23), we obtain respec-tively the positivity conditions;

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2

2

0 e 0, 0 e 0,

0 e 0, 0 e 0,

0 e 0, 0 e 0.

h h

h h h

h h

g t k k th h h h

t thn hn h h

t th h

e t e i t i

i t i t t t

r t r z t z

δ δ

ω δ γ δ

µ ρ δ δ α

− + − + +

− + − +

− + + − +

≥ ≥ ≥ ≥

≥ ≥ ≥ ≥

≥ ≥ ≥ ≥

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Similarly for the modified mosquito population, equations (20)-(27) gives the positivity conditions;

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )0 e 0, 0 e 0,

0 e 0, 0 e 0.

m m h m

m m

m t i tm m m m

t tm m m m

l t l s t s

e t e i t i

δ β δ

φ δ δ

− + − +

− + −

≥ ≥ ≥ ≥

≥ ≥ ≥ ≥

2.11. Existence and Stability of Steady-State Solutions

Let ( )0 0 0 0 0 0 0 0 0 0 0 0, , , , , , , , , ,h h h hn h h m m m mE s e i i t z r l s e i= be the steady-state of the system (17)-(27) which can be calculated by setting the right hand side of the model (17)-(27) to zero, giving us the following;

( ) 2 11 0 h h h m h h h hq z r i s s sα ρ β α δ− Λ + + − − − = (28)

0h m h h h hi s ge eβ δ− − = (29)

1 2 0h h h h h hge q k i k i iδ+ Λ − − − = (30)

( )2 0h h h hnk i iω δ− + = (31)

1 0h h h hk i t tγ δ− − = (32)

( ) 0h h ht rγ µ ρ δ− + + = (33)

1 2 0h h hs r z zα µ α δ+ − − = (34)

0m m m mml lδΛ − − = (35)

0m m h m m mml i s sβ δ− − = (36)

0m h m m m mi s e eβ φ δ− − = (37)

0m m me iφ δ− = (38)

2.12. Disease-Free Equilibrium Point

Disease-free equilibrium points (DFE) are steady-state solutions where there is no disease (malaria). The disease free equilibrium of the normalized model (17)- (27) is obtained by setting

d d d d d d d d d dd 0d d d d d d d d d d d

h h h hn h h m m m ms e i i t r l s e izt t t t t t t t t t t= = = = = = = = = = =

At disease free equilibrium we have,

( )

, ,

0.

h mh m

h m m

h h hn h h m m m

ms sm

e i i t r l e i z qδ δ δΛ Λ

= =+

= = = = = = = = = =

Therefore the disease free equilibrium (DFE) denoted by 0E of the system (28)-(38) is given by

( )

( )

0 0 0 0 0 0 0 0 0 0 0 0, , , , , , , , , ,

, 0, 0,0,0,0,0,0, , 0,0

h h h hn h h m m m m

h m

h m m

E s e i i t z r l s e i

mmδ δ δ

=

Λ Λ= +

that represents the state in which there is no infection in the society and is known as the disease-free equilibrium point (DFE). This implies that at the dis-

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ease-free equilibrium, the susceptible human population is equal to the total human population and the susceptible mosquito population is equal to the total mosquito population.

2.13. Local Stability of DFE

The disease free equilibrium of the model (17)-(27) was given by

( )

( )

0 0 0 0 0 0 0 0 0 0 0 0, , , , , , , , , ,

, 0, 0,0,0,0,0,0, , 0,0

h h h hn h h m m m m

h m

h m m

E s e i i t z r l s e i

mmδ δ δ

=

Λ Λ= +

2.14. Basic Reproduction Ratio

R0 is often found through the study and computation of the eigenvalues of the Jacobian at the disease- or infectious-free equilibrium Diekmann [12] follow a different approach which is the next generation matrix method. This procedure converts a system of ordinary differential equations of a model of infectious dis-ease dynamics to an operator (or matrix) that translate from one generation of infectious individuals to the next. The basic reproductive number is then defined as the spectral radius (dominant eigenvalue) of this operator. Van den Driessche and Watmough [9] describe such a method in detail for general deterministic compartmental models.

The dynamics of the model is specified by the IVP;

( ) ( )d , 0d

nii

x f x xt += ∈ (39)

We define 0Θ as the set of all disease-free states as

0 : 0,1nix x i m+Θ = ∈ = ≤ ≤ (40)

Next we recast the IVP (4.39) in the form;

( ) ( )dd

ii i

x F x V xt= − (41)

where ( )iF x is the rate of new infections entering compartment i, and

( ) ( )i i iV V x V x− += − (42)

where ( )iV x+ is the rate of transfer into compartment i by any other means, and ( )iV x− is the rate of transfer out of compartment i. Given a disease-free equilibrium point DFEx of (39), with DFEx and ( )f x satisfying certain im-portant assumptions [12], then we define the square matrices F and V of dimen-sion m m× as follows;

( ) ( ), for 1 ,DFE DFE

i iij ij

j jx x

F x V xF V i j m

x x∂ ∂

= = ≤ ≤∂ ∂

(43)

It then follows that 1FV − is the next generation matrix and the basic repro-duction ratio 0R is the spectral radius of 1FV − ,

( )10 R FVρ −⇒ = (44)

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Rewriting the system (41) starting with the infected compartments for both populations; , , , , ,h h m m hn he i e i i t and then followed by uninfected classes;

, , , ,h h m ms z r l s also from the two populations, gives;

dd

hh m h h h h

e i s ge et

β δ= − −

1 2dd

hh h h h h h

i ge q k i k i it

δ= + Λ − − −

dd

mm h m m m m

e i s e et

β φ δ= − −

dd

mm m m

i e it

φ δ= −

( )2dd

hnh h h hn

i k i it

ω δ= − +

1dd

hh h h h

t k i t tt

γ δ= − −

( ) 2 1d 1d

hh h h m h h h h

s q z r i s s st

α ρ β α δ= − Λ + + − − −

1 2dd h h hz s r z zt

α µ α δ= + − −

( )dd

hh h h

r t rt

γ µ ρ δ= − + +

dd

mm m m m

l ml lt

δ= Λ − −

dd

mm m h m m m

s ml i s st

β δ= − −

The method of next generation matrix has been used to show the rate of ap-pearance of new infection in compartments; he and me , from the system (12);

( )( )

( )

( )( )

1 2

2

1

0

,000

h hh m h

h h h h

m mm h m

m m m

h h h hn

h h h

g ei sge q k k i

ei sF V

e ik i ik i t

δβδ

φ δβφ δ

ω δγ δ

+ − − Λ + + + + = =

− + − + + − + +

By linearization approach, the associated matrix at disease free equilibrium is obtained as

( )

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

h h

h

m m

m m

mF m

βδ

βδ δ

Λ

Λ = +

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DOI: 10.4236/am.2018.911081 1252 Applied Mathematics

( )

( ) ( ) ( ) ( )1

1 2 1 2

0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

h h h h

h m m h m

m m m m

m m h h m m h

m g mFV

m g k k m k k

β φ βδ δ φ δ δ δ

β βδ δ δ δ δ δ δ

−

Λ Λ +

Λ Λ = + + + + + + +

( ) ( ) ( ) ( )01 2

h h m m

h m m m m h h

m gRm g k k

β φ βδ δ φ δ δ δ δ δ Λ Λ

∴ = + + + + +

Here the term ( )h h

h m m

β φδ δ φ δ

Λ+

explains the number of humans that one mos-

quito infect through contact during the life time it survives as infectious. On the

other hand ( ) ( ) ( )1 2

m m

m m h h

m gm g k k

βδ δ δ δ

Λ+ + + +

describes the number of mos-

quitoes that are infected through contacts with the infectious human during in-fectious period. Hence

0 0 0m hR R R= ×

where ( )0h h

mh m m

R β φδ δ φ δ

Λ=

+ and

( ) ( ) ( )01 2

m mh

m m h h

m gRm g k k

βδ δ δ δ

Λ=

+ + + +.

3. Sensitivity Analysis of the Model Parameters

In this section, we carry out the sensitivity analysis of the model parameter to help us know the parameters that have high impact on the disease transmission, which is on the reproduction ratio 0R .

We used the normalized forward sensitivity index of a variable to parameter approach used in Okosun [13].

3.1. Sensitivity Analysis of R0

We compute the sensitivity of 0R to each of the parameters described in Table 4. Using the formula

mn

m nn m

γ ∂= ×∂

where n represents the variables of the model, and m the parameters.

Sensitivity index of φ given by 12 m

φφ δ

− +

Table 4. Sensitivity index of parameters.

Parameter Hβ 1k 2k g Hδ M Mβ Mδ φ

Sensitivity Index 0.5 –0.25 –0.156 –0.375 –0.719 0.125 0.5 –1.352 –0.272

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Sensitivity index of g given by 12 h

gg δ

− +

Sensitivity index of mδ given by 1 22

m m

m mmδ δ

φ δ δ − − − + +

Sensitivity index of hδ given by 1 2

1 12

h h

h hg k kδ δδ δ

− − − + + +

Sensitivity index of 1k given by 1

1 2

12 h

kk k δ

− + +

Sensitivity index of 2k given by 2

1 2

12 h

kk k δ

− + +

Sensitivity index of m given by 1 12 m

mm δ

− +

Sensitivity index of 12m m h hβ βΛ = = = Λ =

Remark: Sensitivity indices of R0 evaluated at the baseline parameter values are given in the Table 5.

From Table 5, the sensitivity index may be a complex expression, depending on different parameters of the system. But it can also be a constant value. Exam-ple, the sensitivity index of Mβ , Hβ = +0.5, means that increasing (or de-creasing) Mβ , Hβ by 10% increases (or decreases) R0 by 5%.

3.2. Math Cad Simulation of the Model

Parameter values:

1 2

1 1 2

1

: 0.1, : 0.5, : 0.8, : 0.6, : 0.02, : 0.5, : 0.3,: 0.9 : 0.8 : 0.5, : 0.5, : 0.7, : 0.4, : 100,

: 0.2, : 0.3, : 0.1, : 0.4, : 0.12

H H H

H H

M M M

qg k k N

m

α α ρ β δω γ µ

δ β φ

= Λ = = = = = == = = = = = =

Λ = = = = =

( )

( )

( )

( )

10 02 6 5 1 0 0

10 01 1 1

1 1 1 1 2 2 2 2

2 2 3

1 2 4 4

4 5

1 0 5 2 6 6

1 7 7

8 21 7 8

8 29 9

9 10

1

, :

HH H

H

HH

H

H H

H H

H

H

H

M M

MM

H

MM

H

M

Y Yq Y Y Y Y

N

Y Yg Y Y

N

g Y q k Y k Y Yk Y Yk Y Y Y

D t Y Y YY Y Y Y

m Y YY Y

m Y YN

Y YY Y

N

Y Y

βα ρ α δ

βδ

δ

ω δ

γ δ

γ µ ρ δ

α µ α δ

δ

βδ

βφ δ

φ δ

− Λ + + − − −

− −

+ Λ − − −

− +

− −

= − + +

+ − −

Λ − −

− −

− −

−

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Vector of derivative values at any solution point (t, Y): Define additional arguments for the ODE solver:

0 : 0t = : Initial value of independent variable 1: 0t = : Initial value of independent variable

[ ]T0 : 50 15 25 2 2 4 2 5 3 2 1Y = : Vector of initial function values

3: 1 10num = × : Number of solution values on [t0, t1] ( )1: Rkadapt 0, 0, 1, ,S Y t t num D= : Solution matrix

Human (Table 6) 0: 1t S= : Independent variable values

1: 1HS S= : First solution function values 2: 1HE S= : Second solution function values

3: 1HI S= : Third solution function values 4: 1HNI S= : Fourth solution function values

5: 1HT S= : Fifth solution function values 6: 1HR S= : Sixth solution function values 7: 1HA S= : Seventh solution function values

Table 5. Sensitivity indices of R0 evaluated at the baseline parameter values.

Param HΛ MΛ ρ Hβ 2α q 1α 1k 2k g γ

Hδ µ M Mβ Mδ φ Hω

Value 0.5 0.4 0.02 0.5 0.6 0.1 0.8 0.8 0.5 0.9 0.7 0.3 0.4 0.3 0.15 0.1 0,12 0.5

Table 6. Solution matrix S1 for the system of ODEs

0 1 2 3 4 5 6 7

0 0 50 15 25 2 2 4 2

1 0.01 49.469 14.824 24.737 2.108 2.178 3.99 2.394

2 0.02 48.946 14.649 24.476 2.214 2.352 3.97 2.78

3 0.03 48.431 14.477 24.218 2.317 2.523 3.96 3.159

4 0.04 47.924 14.307 23.963 2.419 2.689 3.95 3.53

5 0.05 47.425 14.138 23.71 2.518 2.852 3.94 3.894

6 0.06 46.933 13.972 23.46 2.616 3.012 3.93 4.25

7 0.07 46.449 13.808 23.212 2.711 3.167 3.93 4.6

8 0.08 45.972 13.645 22.966 2.804 3.32 3.92 4.942

9 0.09 45.503 13.485 22.723 2.896 3.469 3.92 5.278

10 0.1 45.041 13,326 22.483 2.985 3.614 3.91 5.607

11 0.11 44.585 13.17 22.245 3.073 3.756 3.91 5.929

12 0.12 44.137 13.015 22.009 3.159 3.895 3.91 6.245

13 0.13 43.695 12.862 21.776 3.242 4.03 3.91 6.554

14 0.14 43.26 12.711 21.545 3.324 4.163 3.91 6.857

15 0.15 42.832 12.561 21.316 3.405 4.292 3.91 …

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Mosquitoes 8: 1ML S= : Eighth solution function values 9: 1MS S= : Ninth solution function values 10: 1ME S= : Tenth solution function values

11: 1MI S= : Eleventh solution function values

3.3. Results and Discussion

The susceptible human population HS against time (Figure 6(a)), clearly shows a rapid exponential decline from the initial value to zero. Similarly, the variation of exposed human population HE against time (Figure 6(b)) depicts an exponential decline from the initial value to zero. The variation of the in-fected human population HI against time (Figure 6(c)), also depicts an expo-nential decline from the initial value to zero. The graphical profile of the varia-tion of the non treated human population HNI against time (Figure 6(d)), shows a sharp rise from the initial value to reach a maximum, and thereafter ex-hibits an exponential decline to zero. The variation of treated human population

HT against time (Figure 6(e)), shows a sharp rise from the initial value to reach a maximum, and thereafter declines exponentially to zero. Similarly, the graphi-cal profile of the variation of the removed human population HR against time (Figure 6(f)), depicts a rise from the initial value to reach a maximum, and the-reafter declines exponentially to zero. The graphical profile of the variation of the protected human population HA against time (Figure 6(g)), shows a sharp rise from the initial value to reach a maximum, and thereafter declines exponen-tially to a steady state. From the graphical profile of the variation of population of mosquito larva ML against time (Figure 6(h)), we observe an exponential decline from the initial value to reach a steady state. The variation of the sus-ceptible mosquito population MS against time (Figure 6(i)), depicts a rise from the initial value to reach a maximum, and thereafter exhibits a sharp de-cline. In the same manner, the variation of the exposed mosquito population

ME against time (Figure 6(j)), shows a decline from the initial value to reach a steady state. Finally, the variation of the infected mosquito population MI against time (Figure 6(k)), depicts a rise from the initial value to reach a maxi-mum and then exhibits a decline.

3.4. Conclusion

Despite the availability of drugs, the malaria disease is still endemic in many parts of the world including developed countries. Elimination of malaria re-quires maintaining the effective reproduction number R0 less than unity, as well as achieving low levels of susceptibility. In this research work, we developed a compartmental bio-mathematical model to study the effect of treatment in the control of malaria in a population with infected immigrants. We obtained the basic reproduction number, R0 and studied the stability of the disease-free equi-librium of the model. Sensitivity analysis of R0 with respect to the model para-meters was carried out on the compartmental vector-host malaria model with

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DOI: 10.4236/am.2018.911081 1256 Applied Mathematics

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

(k)

Figure 6. (a) Population of susceptible humans against time; (b) Population of exposed humans against time; (c) Population of infected humans against time; (d) Population of non-treated infected human against time; (e) Population of treated humans against time; (f) Population of recovered humans against time; (g) Population of protected humans against time; (h) Population of mosquitoes larva against time; (i) Population of suscepti-ble mosquitoes against time; (j) Population of exposed mosquitoes against time; (k) Pop-ulation of infected mosquitoes against time.

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eleven compartments. From the literature on modelling of vector-host malaria models, we discovered that many researchers failed to consider protective meas-ures in their models, though some discussed it theoretically. Our major contri-bution to the existing body of knowledge is incorporating the protective measure in our mathematical model.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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